statsmodels.tsa.statespace.tools.constrain_stationary_multivariate¶

statsmodels.tsa.statespace.tools.constrain_stationary_multivariate(unconstrained, variance, transform_variance=False, prefix=None)[source]¶

Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation for a vector autoregression.

Parameters:¶

unconstrainedarray or list: Arbitrary matrices to be transformed to stationary coefficient matrices of the VAR. If a list, should be a list of length order, where each element is an array sized k_endog x k_endog. If an array, should be the matrices horizontally concatenated and sized k_endog x k_endog * order.
error_variancendarray: The variance / covariance matrix of the error term. Should be sized k_endog x k_endog. This is used as input in the algorithm even if is not transformed by it (when transform_variance is False). The error term variance is required input when transformation is used either to force an autoregressive component to be stationary or to force a moving average component to be invertible.
transform_variancebool, optional: Whether or not to transform the error variance term. This option is not typically used, and the default is False.
prefix{‘s’,’d’,’c’,’z’}, optional: The appropriate BLAS prefix to use for the passed datatypes. Only use if absolutely sure that the prefix is correct or an error will result.

Returns:¶

constrainedarray or list: Transformed coefficient matrices leading to a stationary VAR representation. Will match the type of the passed unconstrained variable (so if a list was passed, a list will be returned).

Notes

In the notation of [1], the arguments (variance, unconstrained) are written as $(Σ, A_{1}, \dots, A_{p})$ , where $p$ is the order of the vector autoregression, and is here determined by the length of the unconstrained argument.

There are two steps in the constraining algorithm.

First, $(A_{1}, \dots, A_{p})$ are transformed into $(P_{1}, \dots, P_{p})$ via Lemma 2.2 of [1].

Second, $(Σ, P_{1}, \dots, P_{p})$ are transformed into $(Σ, ϕ_{1}, \dots, ϕ_{p})$ via Lemmas 2.1 and 2.3 of [1].

If transform_variance=True, then only Lemma 2.1 is applied in the second step.

While this function can be used even in the univariate case, it is much slower, so in that case constrain_stationary_univariate is preferred.

References

[1] (1,2,3)

Ansley, Craig F., and Robert Kohn. 1986. “A Note on Reparameterizing a Vector Autoregressive Moving Average Model to Enforce Stationarity.” Journal of Statistical Computation and Simulation 24 (2): 99-106.